Continuous selections of multivalued mappings

331 Pages·1997·1.503 MB·English

by  Repovs D., Semenov P.V.

Contents р. 1из 4
Contents

Preface
A. Theory
0. Preliminaries
1. Topological spaces
2. Topological vector spaces
3. Banach spaces
4. Extensions of continuous functions
5. Multivalued mappings
1. Convex-valued selection theorem
1. Paracompactness of the domain as a necessary
condition
2. The method of outside approximations
3. The method of inside approximations
4. Properties of paracompact spaces
5. Nerves of locally finite coverings
6. Some properties of paracompact spaces
2. Zero-demensional selection theorem
1. Zero-dimensionality of the domain as a necessary
condition
2. Proof of Zero-dimensional selection theorem
3. Relations between Zero-dimensional and Convex-valued selection theorems
1. Preliminaries. Probabilistic measure and integration
2. Milyutin mappings. Convex-valued selection theorem
via Zero-dimensional theorem

3. Existence of Milyutin mappings on the class of
paracompact spaces
4. Zero-dimensionality of X and continuity of f
0
4. Compact-valued selection theorem
1. Approach via Zero-dimensional theorem
2. Proof via inside approximations
3. Method of coverings
5. Finite-dimensional selection theorem
n n
1. C and LC subsets of topological spaces
2. Shift selection theorem. Sketch of the proof
3. Proofs of main lemmas and Controlled contractibility
theorem
4. From Nerve-weak shift selection theorem to Shift
selection theorem
5. From Controlled extension theorem to Nerve-weak
shift selection theorem
Contents . 2из 4
6. Controlled extension theorem
7. Finite-dimensional selection theorem. Uniform relative
version
n n
8. From UELC restrictions to ELC restrictions
6. Examples and counterexamples
7. Addendum New proof of Finite-dimensional selection theorem
1. Filtered multivalued mappings. Statements of the
results
2. Singlevalued approximations of upper semicontinuous
mappings

3. Separations of multivalued mappings. Proof of
Theorem (17.6)
4. Enlargements of compact-valued mappings. Proof of
Theorem (17.7)
B. Results
1. Characterization of normality-type properties
1. Some other convex-valued selection theorems
2. Characterizations via compact-valued selection
theorems

3. Dense families of selections. Characterization of
perfect normality
n
4. Selections of nonclosed-valued equi-LC mappings
2. Unified selection theorems
1. Union of Finite-dimensional and Convex-valued theorems.
Approxi-mative selection properties

2. Countable type selection theorems and their unions with
other selection theorems
3. Selection theorems for non-lower semicontinuous mappings
1. Lower semicontinuous selections and derived mappings
2. Almost lower semicontinuity
3. Quasi lower semicontinuity
4. Further generalizations of lower semicontinuity
5. Examples
4. Selection theorems for nonconvex-valued maps
1. Paraconvexity. Function of non-convexity of closed subsets of normed spac
2. Axiomatic definition of convex structures in metric spaces. Geodesic
structures
3. Topological convex structure
4. Decomposable subsets of spaces of measurable functions
5. Miscellaneous results
1. Metrizability of the range of a multivalued mapping
2. A weakening of the metrizability of the ranges
Contents р. 3из 4
3. Hyperspaces, selections and orderability
4. Densely defined selections
5. Continuous multivalued approximations of semicontinuous multival-ued
mappings
6. Various results on selections
7. Recent results
6. Measurable selections
1. Uniformization problem
2. Measurable multivalued mappings
3. Measurable selections of semicontinuous mappings
4. Caratheodory conditions. Solutions of differential inclusions
C. Applications
1. First applications
1. Extension theorems
2. Bartle-Graves type theorems. Theory of liftings
3. Homeomorphism problem for separable Banach spaces
4. Applications of Zero-dimensional selection theorem
5. Continuous choice in continuity type definitions
6. Paracompactness of CW-complexes
7. Miscellaneous results
2. Regular mappings and locally trivial fibrations
1. Dyer-Hamstrom theorem
2. Regular mappings with fibers homeomorphic to the interval
3. Strongly regular mappings
4. Noncompact fibers. Exact Milyutin mappings
3. Fixed-point theorems
1. Fixed-point theorems and fixed-point sets for convex-valued mappings
2. Fixed-point sets of nonconvex valued mappings
3. Hilbert space case
4. An application of selections in the finite-dimensional case
5. Fixed-point theorem for decomposable-valued contractions
4. Homeomorphism Group Problem
1. Statement of the problem. Solution for n = 1
2. The space of all self-homeomorphisms of the disk
5. Soft mappings
1. Dugundji spaces and AE(0)-compacta
2. Dugundji mappings and 0-soft mappings
3. General concept of softness. Adequacy problem
4. Parametric versions of Vietoris-Wazewski-Wojdyslawski theorem
5. Functor of probabilistic measures
6. Metric projections
1. Proximinal and Čebyšev subsets of normed spaces
Contents р. 4из 4
2. Continuity of metric projections and projections
3. Continuous selections of metric projections in spaces of continuous function
and L -spaces
p
4. Rational approximations in spaces of continuous functions and and L -spac
p
7. Differential inclusions
1. Decomposable sets in functional spaces
2. Selection approach to differential inclusions. Preliminary results
3. Selection theorems for decomposable valued mappings
4. Directionally continuous selections
References
Subject Index
(cid:2)
Preface
This book is dedicated to the theory of continuous selections of multi(cid:8)
valued mappings(cid:18) a classical area of mathematics (cid:15)as far as the formulation
of its fundamentalproblemsand methodsof solutions are concerned(cid:16) as well
as an area which has been intensively developing in recent decades and has
found various applications in general topology(cid:18) theory of absolute retracts
and in(cid:13)nite(cid:8)dimensional manifolds(cid:18) geometric topology(cid:18) (cid:13)xed(cid:8)point theory(cid:18)
functional and convex analysis(cid:18) game theory(cid:18) mathematical economics(cid:18) and
other branches of modernmathematics(cid:3) The fundamentalresults in this the(cid:8)
ory were laid down in the mid (cid:0)(cid:9)(cid:1)(cid:2)(cid:19)s by E(cid:3) Michael(cid:3)
The book consists of (cid:15)relatively independent(cid:16) three parts (cid:20) Part A(cid:12)
Theory(cid:18) Part B(cid:12) Results(cid:18) and Part C(cid:12) Applications(cid:3) (cid:15)We shall refer to these
parts simply by their names(cid:16)(cid:3) The target audience for the (cid:13)rst part are
studentsofmathematics(cid:15)intheirsenioryear orintheir(cid:13)rstyear ofgraduate
school(cid:16) who wish to get familiar with the foundations of this theory(cid:3) The
goal of the second part is to give a comprehensive survey of the existing
results on continuous selections of multivalued mappings(cid:3) It is intended for
specialists in this area as well as for those who have masteredthe materialof
the (cid:13)rst part of the book(cid:3) In the third part we present important examples
of applications of continuous selections(cid:3) We have chosen examples which
are su(cid:21)ciently interesting and have played in some sense key role in the
corresponding areas of mathematics(cid:3) The necessary prerequisites can all be
foundin the(cid:13)rst part(cid:3) It is intendedforresearchers ingeneral andgeometric
topology(cid:18) functional and convex analysis(cid:18) approximation theory and (cid:13)xed(cid:8)
(cid:8)point theory(cid:18) di(cid:17)erential inclusions(cid:18) and mathematical economics(cid:3)
The style of exposition changes as we pass from one part of the book
to another(cid:3) Proofs in Theory are given in details(cid:3) Here(cid:18) our philosophy was
to present (cid:22)the minimumof facts with the maximumof proofs(cid:23)(cid:3) In Results(cid:18)
however(cid:18) proofs are(cid:18) as a rule(cid:18) omitted or are only sketched(cid:3) In other words(cid:18)
as it is usual for advanced expositions(cid:18) we give here (cid:22)the maximumof facts
with the minimum of proofs(cid:23)(cid:3) Finally(cid:18) in every paragraph of Applications
the part concerning selections is studied in details whereas the rests of the
argument is usually only sketched(cid:3) So the style is of mixed type(cid:3)
Next(cid:18) we wish to explain the methodical approach in Theory(cid:3) We have
presented the proofs in some (cid:13)xed structurized form(cid:3) More precisely(cid:18) every
theoremisprovedintwosteps(cid:24) PartI(cid:12)ConstructionandPartII(cid:12)Veri(cid:0)cation(cid:3)
The (cid:13)rst part lists all steps of the proof and in the sequel we formulate the
necessary properties of the construction(cid:3) The second part brings a detailed
(cid:0)
(cid:5)
veri(cid:13)cation of each of the statements of the (cid:13)rst part(cid:3) In this way(cid:18) an
experienced reader can only browse through the (cid:13)rst part and then skip the
second part altogether(cid:18) whereas a beginner may well wish to pause after
Construction andtry to verify all steps by himself(cid:3) Inthis way(cid:18) Construction
part can also be regarded as a set of exercises on selection theory(cid:3)
Consequently(cid:18) there are no special exercise sections in Theory after each
paragraph(cid:12) instead(cid:18) we have organized each proof as a sequence of exercises(cid:3)
WehavealsoprovidedTheorywithaseparateintroduction(cid:18)whereweexplain
thewaysinwhichmultivaluedmappingsandtheircontinuousselections arise
in di(cid:17)erent areas of mathematics(cid:3)
Some comments concerning terminology and notations(cid:12) A multivalued
mappingtoaspaceY canbede(cid:13)nedasasinglevaluedmappingintoasuitable
spaceofsubsetsofY(cid:3) Suchapproachforcesusto introduceaspecialnotation
for speci(cid:13)c classes of subsets of Y(cid:3) In the following table we have collected
various notations which one can (cid:13)nd in the literature(cid:12)
Classes of subsets of Y Notations
Y
all subsets A(cid:11)Y(cid:12)(cid:0) (cid:5) (cid:0) P(cid:11)Y(cid:12)(cid:0) B(cid:11)Y(cid:12)
Y
all nonempty subsets (cid:5) (cid:0) expY(cid:0) N(cid:11)Y(cid:12)
closed expY(cid:0) F(cid:11)Y(cid:12)(cid:0) Cl(cid:11)Y(cid:12)(cid:0) C(cid:11)Y(cid:12)
compact Cp(cid:11)Y(cid:12)(cid:0) C(cid:11)Y(cid:12)(cid:0) K(cid:11)Y(cid:12)(cid:0) Comp(cid:11)Y(cid:12)(cid:0) C(cid:11)Y(cid:12)
(cid:10)nite F(cid:11)Y(cid:12)(cid:0) K(cid:11)Y(cid:12)(cid:0) exp(cid:0)(cid:11)Y(cid:12)(cid:0) J(cid:11)Y(cid:12)
convex Cv(cid:11)Y(cid:12)(cid:0) C(cid:11)Y(cid:12)(cid:0) K(cid:11)Y(cid:12)(cid:0) Conv(cid:11)Y(cid:12)
closed convex FC(cid:11)Y(cid:12)(cid:0)CC(cid:11)Y(cid:12)(cid:0) CC(cid:11)Y(cid:12)(cid:0) CK(cid:11)Y (cid:12)
compact convex Kv(cid:11)Y(cid:12)(cid:0) CK(cid:11)Y(cid:12)(cid:0) ComC(cid:11)Y(cid:12)(cid:0) Uk(cid:11)Y(cid:12)(cid:0) conv(cid:18)(cid:11)Y(cid:12)
complete CMP(cid:11)Y(cid:12)(cid:0) (cid:19)(cid:11)Y(cid:12)
bounded B(cid:11)Y(cid:12)(cid:0) Bd(cid:11)Y(cid:12)
combination of above BdF(cid:11)Y(cid:12)(cid:0) (cid:19)K(cid:11)Y(cid:12)(cid:0) (cid:19)CK(cid:11)Y(cid:12)(cid:0)(cid:1)(cid:1)(cid:1)
We have solved the problem of the choice of notations in a very simple
way(cid:12) we did not make any choice(cid:3) More precisely(cid:18) we prefer the language
instead of abbreviations andwe always (cid:15)except insomeplaces inResults(cid:16) use
phrases of the type (cid:22)let F (cid:12) X Y be a multivalued mapping with closed
(cid:0)
(cid:15)compact(cid:18) bounded(cid:18) etc(cid:3)(cid:16) values(cid:3)(cid:3)(cid:3)(cid:23)(cid:3) The only general agreement is that all
values of any multivalued mapping F (cid:12) X Y are nonempty subsets of Y(cid:2)
(cid:0)
According to our decision(cid:18) we systematically use the notation (cid:22)F (cid:12)X
(cid:0)
Y(cid:23) and associate with it the term (cid:22)multivalued mapping(cid:23)(cid:18) although from
purely pedagogical point of view the last term should be related to notions
Y
of the type (cid:22)F (cid:12) X (cid:4) (cid:18) F (cid:12) X C(cid:15)Y(cid:16)(cid:18) etc(cid:3)(cid:23) Finally(cid:18) a word about
(cid:0) (cid:0) F
cross(cid:8)references in our book(cid:12) when we are e(cid:3)g(cid:3) in Part B(cid:12) Results and refer
to say(cid:18) Theorem (cid:15)A(cid:3)(cid:5)(cid:3)(cid:9)(cid:16) (cid:15)resp(cid:3) De(cid:13)nition (cid:15)C(cid:3)(cid:6)(cid:3)(cid:0)(cid:16)(cid:16)(cid:18) we mean Theorem (cid:15)(cid:5)(cid:3)(cid:9)(cid:16)
of Part A(cid:12) Theory (cid:15)resp(cid:3) De(cid:13)nition (cid:15)(cid:6)(cid:3)(cid:0)(cid:16) of Part C(cid:12) Applications(cid:16)(cid:3)
(cid:1)
(cid:17)
Weconcludebysomecommentsconcerningtheexistingliterature(cid:3) There
already are some textbooks and monographs where some attention is also
given to certain aspects of the theory of selections (cid:25)(cid:0)(cid:10)(cid:18)(cid:0)(cid:6)(cid:18)(cid:4)(cid:1)(cid:18)(cid:5)(cid:2)(cid:18)(cid:0)(cid:5)(cid:0)(cid:18)(cid:4)(cid:11)(cid:2)(cid:18)(cid:4)(cid:9)(cid:11)(cid:18)
(cid:5)(cid:4)(cid:10)(cid:18)(cid:7)(cid:2)(cid:1)(cid:26)(cid:3) However(cid:18) none of them contains a systematic treatment of the
theory and so to the best of our knowledge(cid:18) the present monograph is the
(cid:13)rst one which is devoted exclusively to this subject(cid:3)
Preliminary versions of the book were read by several of our col(cid:8)
leagues(cid:3) In particular(cid:18) we acknowledge remarks by S(cid:3) M(cid:3) Ageev(cid:18) V(cid:3) Gutev(cid:18)
S(cid:3) V(cid:3) Konyagin(cid:18) V(cid:3) I(cid:3) Levin(cid:18) and E(cid:3) Michael(cid:3) The manuscript was prepared
using TEX by M(cid:3) Zemlji(cid:27)c and we are very grateful for his technical help and
assistance throughalltheseyears(cid:3) The(cid:13)rstauthoracknowledges thesupport
ofthe MinistryforScience andTechnology of theRepublicofSlovenia grants
No(cid:3)P(cid:0)(cid:8)(cid:2)(cid:4)(cid:0)(cid:7)(cid:8)(cid:0)(cid:2)(cid:0)(cid:8)(cid:9)(cid:4) andNo(cid:3)J(cid:0)(cid:8)(cid:6)(cid:2)(cid:5)(cid:9)(cid:8)(cid:2)(cid:0)(cid:2)(cid:0)(cid:8)(cid:9)(cid:1)(cid:18) andthesecondauthorthesup(cid:8)
portof theInternationalScienceG(cid:3)SorosFoundationgrantNo(cid:3)NFU(cid:2)(cid:2)(cid:2) and
the Russian Basic Research Foundation grant No(cid:3) (cid:9)(cid:10)(cid:8)(cid:2)(cid:0)(cid:8)(cid:2)(cid:0)(cid:0)(cid:10)(cid:10)a(cid:3)
D(cid:2) Repov(cid:16)s and P(cid:2) V(cid:2) Semenov
(cid:2)
x(cid:2)(cid:1) CONVEX(cid:3)VALUED SELECTION THEOREM
This chapter deals more or less with a single theorem (cid:20) the one stated
in the title(cid:3) This theorem gives su(cid:22)cient conditions for the solvability of
the continuous selection problem for a paracompact domain(cid:3) But in order
to introduce paracompactness from a selection point of view we start by
searching for the necessary condition for the existence of such a solution(cid:3)
In Section (cid:0) we prove that the existence of continuous selections of lower
semicontinuous mappings with closed convex values implies the existence of
locally (cid:13)nite re(cid:13)nements and locally (cid:13)nite partitions of unity(cid:3) In Sections
(cid:4) and (cid:5) we present two approaches to proving the Convex(cid:8)valued selection
theorem(cid:3) In Section (cid:4) the answer is given as a uniform limit of continuous
(cid:1)n(cid:8)selections(cid:18)whereasinSection(cid:5)(cid:18)itisgivenasauniformlimitof(cid:1)n(cid:8)continu(cid:8)
ousselections(cid:3) In(cid:15)auxiliary(cid:16)Section(cid:7)we provetheequivalence ofde(cid:13)nitions
of paracompact spaces via coverings and via partitions of unity(cid:3) Also(cid:18) we
collect there the material concerning the properties of paracompact spaces(cid:18)
nerves of coverings and some facts about dimension theory(cid:3)
All material of this chapter is classical and well(cid:8)known(cid:3) In the proof
of Theorem (cid:15)(cid:0)(cid:3)(cid:0)(cid:16) we follow (cid:25)(cid:5)(cid:2)(cid:26)(cid:18) with a supplement as in (cid:15)(cid:0)(cid:3)(cid:0)(cid:16)$(cid:3) Proof of
Theorem(cid:15)(cid:0)(cid:3)(cid:1)(cid:16)(cid:18) given in Section (cid:4)(cid:18) (cid:13)rst appeared in (cid:25)(cid:4)(cid:1)(cid:6)(cid:18)(cid:4)(cid:1)(cid:11)(cid:26)(cid:3) This approach
was repeated in several textbooks and monographs(cid:3) On the other hand(cid:18) the
second proof of Theorem (cid:15)(cid:0)(cid:3)(cid:1)(cid:16)(cid:18) given in Section (cid:5)(cid:18) is practically unknown(cid:3)
We know of only one article (cid:25)(cid:4)(cid:10)(cid:0)(cid:26) where such an idea was realized(cid:18) however(cid:18)
inamuchmoreabstract situation(cid:3) So(cid:18) perhapsthisproofofTheorem(cid:15)(cid:0)(cid:3)(cid:1)(cid:16) is
new(cid:3) We omitall references in Section (cid:7)(cid:3) They can befoundin any standard
book on general topology(cid:18) e(cid:3)g(cid:3) in (cid:25)(cid:0)(cid:2)(cid:11)(cid:18)(cid:0)(cid:0)(cid:11)(cid:26)(cid:3)
(cid:0)(cid:1) Paracompactness of the domain as a necessary condition
Theorem (cid:2)(cid:4)(cid:0)(cid:4)(cid:5)(cid:3) Let X be a topological space such that each lower semi(cid:1)
continuous map from X into any Banach space with closed convex values
admits a continuous singlevalued selection(cid:2) Then every open covering of X
admits a locally (cid:0)nite open re(cid:0)nement(cid:2)
Proof(cid:2)
I(cid:2) Construction
Let(cid:12)
(cid:15)(cid:0)(cid:16) (cid:2) (cid:14) G(cid:1) (cid:1)(cid:0)A be an open covering of the space X(cid:24)
f g
(cid:15)(cid:4)(cid:16) B (cid:14) l(cid:2)(cid:15)A(cid:16) be the Banach space of all summable functions s (cid:12) A IR
(cid:0)
over the index set A (cid:15)see (cid:2)(cid:3)(cid:5)(cid:16)(cid:24) and
x
(cid:15)(cid:5)(cid:16) For every x X(cid:18) let
(cid:2)
F(cid:15)x(cid:16) (cid:14) s B s (cid:2)(cid:3) s (cid:14) (cid:0) and s(cid:15)(cid:5)(cid:16) (cid:14)(cid:2)(cid:3) whenever x (cid:6) G(cid:1) (cid:0)
f (cid:2) j (cid:12) k k (cid:2) g
(cid:2)(cid:2)
(cid:17)(cid:14) Convex(cid:1)valued selection theorem
We claim that then(cid:12)
(cid:15)a(cid:16) F(cid:15)x(cid:16) is a nonempty convex closed subset of the Banach space B(cid:18) for
every x X(cid:24) and
(cid:2)
(cid:15)b(cid:16) The mapF (cid:12) X B is lower semicontinuous(cid:18) i(cid:3)e(cid:3) for every x X(cid:18) every
(cid:0) (cid:2)(cid:2) (cid:2)
s F(cid:15)x(cid:16) and every (cid:1) (cid:9) (cid:2)(cid:18) the preimage F (cid:15)D(cid:15)s(cid:3)(cid:1)(cid:16)(cid:16) contains an open
(cid:2)
neighborhood of x(cid:3)
Itfollowsbythehypothesesofthetheoremthat thereexistsacontinuous
selection for F(cid:18) say f(cid:3) Let(cid:12)
(cid:15)(cid:7)(cid:16) e(cid:1)(cid:15)x(cid:16) (cid:14)(cid:25)f(cid:15)x(cid:16)(cid:26)(cid:15)(cid:5)(cid:16)(cid:24)
(cid:15)(cid:1)(cid:16) e(cid:15)x(cid:16) (cid:14)sup e(cid:1)(cid:15)x(cid:16) (cid:5) A (cid:24) and
f j (cid:2) g
(cid:15)(cid:10)(cid:16) V(cid:1) (cid:14) x X e(cid:1)(cid:15)x(cid:16) (cid:9)e(cid:15)x(cid:16)(cid:6)(cid:4) (cid:3)
f (cid:2) j g
We claim that then(cid:12)
(cid:15)c(cid:16) e(cid:1) is a continuous function from X into (cid:25)(cid:2)(cid:3)(cid:0)(cid:26) and (cid:1)(cid:0)Ae(cid:1)(cid:15)x(cid:16) (cid:14) (cid:0)(cid:18) for
every x X(cid:24)
(cid:2) P
(cid:15)d(cid:16) e is a continuous positive function(cid:24)
(cid:15)e(cid:16) If e(cid:1)(cid:15)x(cid:16) (cid:9)(cid:2) then x G(cid:1)(cid:18) for all (cid:5) A(cid:24)
(cid:2) (cid:2)
(cid:15)f(cid:16) V(cid:1) G(cid:1)(cid:18) for all (cid:5) A(cid:24)
(cid:5) (cid:2)
(cid:15)g(cid:16) V(cid:1) is a locally (cid:13)nite family of open subsets of the space X(cid:24) and
f g
(cid:15)h(cid:16) The family V(cid:1) (cid:1)(cid:0)A(cid:18) is a cover of the space X(cid:3)
f g
II(cid:2) Veri(cid:0)cation
(cid:15)a(cid:16) Let A(cid:15)x(cid:16) (cid:14) (cid:5) A x G(cid:1) (cid:3) It is easy to see that F(cid:15)x(cid:16) is the standard
f (cid:2) j (cid:2) g
basic simplex in the Banach space l(cid:2)(cid:15)A(cid:15)x(cid:16)(cid:16)(cid:18) see (cid:2)(cid:3)(cid:5)(cid:3)
x
(cid:15)b(cid:16) For x X(cid:18) s F(cid:15)x(cid:16) and (cid:1) (cid:9) (cid:2)(cid:18) let us (cid:13)rst consider the case when
(cid:2) (cid:2)
supp(cid:15)s(cid:16) (cid:14) (cid:5) A s(cid:15)(cid:5)(cid:16) (cid:9) (cid:2) (cid:14) (cid:5)(cid:2)(cid:3)(cid:5)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:5)N is a (cid:0)nite subset of A(cid:3)
f (cid:2) j g f g (cid:1)
Thendueto theconstructionofthemappingF(cid:18)thepointsbelongsto F(cid:15)x(cid:16)(cid:18)
(cid:1)
for every x from the neighborhood G(cid:15)x(cid:16) (cid:14) i(cid:8)N G(cid:1)i of the point x(cid:3) Hence
(cid:2)(cid:2) (cid:2)(cid:2)
G(cid:15)x(cid:16) F (cid:15) s (cid:16) F (cid:15)D(cid:15)s(cid:3)(cid:1)(cid:16)(cid:16)(cid:18) i(cid:3)e(cid:3) F is lower semicontinuous at x(cid:3) The
(cid:5) f g (cid:5) T
second case of countable supp(cid:15)s(cid:16) follows from the (cid:13)rst case and from the
obvious fact that in the standard simplexof the space (cid:14)(cid:2) the subset of points
with (cid:13)nite supports constitutes a dense subset(cid:3)
(cid:15)c(cid:16) The function e(cid:1) (cid:12)X (cid:25)(cid:2)(cid:3)(cid:0)(cid:26) is a composition of the continuous selection
(cid:0)
f and the (cid:22)(cid:5)(cid:8)th coordinate(cid:23) projection p(cid:1) of the entire Banach space l(cid:2)(cid:15)A(cid:16)(cid:3)
The equality (cid:1)(cid:0)Ae(cid:1)(cid:15)x(cid:16) (cid:14)(cid:0) follows by (cid:15)(cid:7)(cid:16) and since f(cid:15)x(cid:16) F(cid:15)x(cid:16)(cid:3)
(cid:2)
(cid:15)d(cid:16) For an arbitrary x X(cid:18) we pick an index (cid:12) (cid:14)(cid:12)(cid:15)x(cid:16) such that e(cid:6)(cid:15)x(cid:16) (cid:9)(cid:2)(cid:3)
P (cid:2)
Then for some (cid:13)nite set of indices (cid:30)(cid:15)x(cid:16) A we have that
(cid:5)
(cid:0) e(cid:1)(cid:15)x(cid:16) (cid:8) e(cid:6)(cid:15)x(cid:16)(cid:6)(cid:4)(cid:0)
(cid:16)
(cid:1)(cid:0)(cid:1)(cid:8)x(cid:9)
X
Onthe left side isthe sumof a (cid:13)nite numberof continuous functions(cid:3) Hence(cid:18)
the inequality
e(cid:1)(cid:15)z(cid:16) (cid:14)(cid:0) e(cid:1)(cid:15)z(cid:16) (cid:8)e(cid:6)(cid:15)z(cid:16)(cid:6)(cid:4)
(cid:16)
(cid:1)(cid:0)(cid:4)(cid:1)(cid:8)x(cid:9) (cid:1)(cid:0)(cid:1)(cid:8)x(cid:9)
X X
(cid:2)(cid:3)
Paracompactness of the domain as a necessary condition (cid:17)(cid:13)
holds for every z from some open neighborhood W(cid:15)x(cid:16) of the point x(cid:3) But
then e(cid:0)(cid:15)z(cid:16) (cid:8) e(cid:6)(cid:15)z(cid:16)(cid:18) for all (cid:2) (cid:6) (cid:30)(cid:15)x(cid:16)(cid:3) So we have proved that the function
(cid:2)
e(cid:15)(cid:16) is in fact the maximumof a (cid:13)nite numberof continuous functions in the
(cid:14)
neighborhood W(cid:15)x(cid:16)(cid:3) Therefore e(cid:15) (cid:16) is continuous(cid:3) Finally(cid:18) positivity of e(cid:15) (cid:16)
(cid:14) (cid:14)
follows from (cid:15)c(cid:16)(cid:3)
(cid:15)e(cid:16) If x (cid:6) G(cid:1) then for every s F(cid:15)x(cid:16)(cid:18) we have that s(cid:15)(cid:5)(cid:16) (cid:14) (cid:2)(cid:18) (cid:15)see (cid:15)(cid:5)(cid:16)(cid:16)(cid:3) So
(cid:2) (cid:2)
by f(cid:15)x(cid:16) F(cid:15)x(cid:16)(cid:18) we get e(cid:1)(cid:15)x(cid:16) (cid:14)(cid:25)f(cid:15)x(cid:16)(cid:26)(cid:15)(cid:5)(cid:16) (cid:14)(cid:2)(cid:3)
(cid:2)
(cid:15)f(cid:16) This follows frome(cid:1)(cid:15)x(cid:16) (cid:9)e(cid:15)x(cid:16)(cid:6)(cid:4) e(cid:6)(cid:15)x(cid:16)(cid:6)(cid:4)(cid:18) from(cid:15)e(cid:16)(cid:18) and frome(cid:6)(cid:15)x(cid:16) (cid:9)
(cid:12)
(cid:9)(cid:2) (cid:15)see the proof of (cid:15)d(cid:16)(cid:16)(cid:3)
(cid:15)g(cid:16) It follows by (cid:15)(cid:10)(cid:16) and by continuity of functions e(cid:1) and e that V(cid:1) is an
open set(cid:3) As in the proof of (cid:15)d(cid:16) we (cid:13)nd for an arbitrary x(cid:18) some (cid:13)nite set
(cid:30)(cid:15)x(cid:16) A and someneighborhoodW(cid:15)x(cid:16) of the point x(cid:3) Then W(cid:15)x(cid:16) V(cid:0) (cid:14)
(cid:5) (cid:3) (cid:13) (cid:1)
holds only for (cid:2) (cid:30)(cid:15)x(cid:16)(cid:3) Indeed(cid:18) if z V(cid:0) W(cid:15)x(cid:16) then
(cid:2) (cid:2) (cid:3)
e(cid:0)(cid:15)z(cid:16) (cid:9) e(cid:15)z(cid:16)(cid:6)(cid:4) e(cid:6)(cid:15)z(cid:16)(cid:6)(cid:4) (cid:9)(cid:0) e(cid:1)(cid:15)z(cid:16) (cid:14) e(cid:1)(cid:15)z(cid:16)(cid:0)
(cid:12) (cid:16)
(cid:1)(cid:0)(cid:1)(cid:8)x(cid:9) (cid:1)(cid:0)(cid:4)(cid:1)(cid:8)x(cid:9)
X X
Hence e(cid:0)(cid:15)z(cid:16) (cid:9)e(cid:1)(cid:15)z(cid:16)(cid:18) for every (cid:5) (cid:6) (cid:30)(cid:15)x(cid:16)(cid:18) i(cid:3)e(cid:3) (cid:2) (cid:30)(cid:15)x(cid:16)(cid:3)
(cid:2) (cid:2)
(cid:15)h(cid:16) Follows by contradiction(cid:12) if x (cid:6) V(cid:1)(cid:18) (cid:5) A(cid:18) then e(cid:1)(cid:15)x(cid:16) e(cid:15)x(cid:16)(cid:6)(cid:4) and
(cid:2) (cid:2) (cid:9)
(cid:2) (cid:8)e(cid:15)x(cid:16) (cid:14) supp e(cid:1)(cid:15)x(cid:16) (cid:5) A e(cid:15)x(cid:16)(cid:6)(cid:4)(cid:3)
f j (cid:2) g (cid:9) S
De(cid:1)nition (cid:2)(cid:4)(cid:0)(cid:6)(cid:5)(cid:3) A Hausdor(cid:17) space X is said to be paracompact if
every open covering of X admits a locally (cid:13)nite open re(cid:13)nement(cid:3)
We can now reformulate Theorem (cid:15)(cid:0)(cid:3)(cid:0)(cid:16) in the following manner(cid:12) Para(cid:1)
compactness of the domain is a necessary condition for existence of contin(cid:1)
uous selections of lower semicontinuous mappings into Banach spaces with
convex closed values(cid:2)
Our (cid:13)rst goal is to prove that paracompactness of the domain is also a
su(cid:21)cient condition for such an existence (cid:15)Sect (cid:4)(cid:16)(cid:3) But here we continue by
ananalogueofTheorem(cid:15)(cid:0)(cid:3)(cid:0)(cid:16) forexistence oflocally(cid:13)nitepartitionsofunity(cid:3)
De(cid:1)nition (cid:2)(cid:4)(cid:0)(cid:7)(cid:5)(cid:3)
(cid:15)a(cid:16) A family e(cid:1) (cid:1)(cid:0)A of nonnegative continuous functions on a topological
f g
space X is said to be a locally (cid:0)nite partition of unity if for every x X(cid:18)
(cid:2)
there exists a neighborhood W(cid:15)x(cid:16) and a (cid:13)nite subset A(cid:15)x(cid:16) A such that
(cid:5)
(cid:1)(cid:0)A(cid:8)x(cid:9)e(cid:1)(cid:15)y(cid:16) (cid:14) (cid:0)(cid:18) for all y W(cid:15)x(cid:16) and e(cid:1)(cid:15)y(cid:16) (cid:14) (cid:2)(cid:18) for y (cid:6) W(cid:15)x(cid:16) and
(cid:2) (cid:2)
(cid:5) A(cid:15)x(cid:16)(cid:3)
P(cid:2)
(cid:15)b(cid:16) A locally (cid:13)nite partition of unity e(cid:1) (cid:1)(cid:0)A is said to be inscribed into an
f g
opencovering G(cid:0) (cid:0)(cid:0)(cid:1) of atopological space X if forany (cid:5) A(cid:18) there exists
f g (cid:2)
(cid:2) (cid:30) such that
(cid:2)
supp(cid:15)e(cid:1)(cid:16) (cid:14) Cl x X e(cid:1)(cid:15)x(cid:16) (cid:9)(cid:2) G(cid:0)(cid:3)
f (cid:2) j g (cid:5)
It is easy to see that for a locally (cid:13)nite partition of unity e(cid:1) (cid:1)(cid:0)A
(cid:2)(cid:2) f g
inscribed into a covering G(cid:0) (cid:0)(cid:0)G(cid:18) the family of open sets e(cid:1) (cid:15)(cid:15)(cid:2)(cid:3)(cid:0)(cid:26)(cid:16) (cid:1)(cid:0)A
f g f g
gives a locally (cid:13)nite re(cid:13)nement of the covering G(cid:0) (cid:0)(cid:0)(cid:1)(cid:3)
f g
(cid:2)(cid:4)